First change the map view to barbarians (or rather, if you use less than 7 civs, the civ that doesn't exist), then choose no human player.
If you don't do the first step, you will still get all the messages meant for the selected civilization, no to mention you'll see lots of units moving around which makes it all incredibly slow too.
So if you change the map view to barbarians first, you won't get any of this... But you will have to move the barbarian units when they pop up.
If you first change the map view to a non-existant civ, you will only get general notices like wonder completion, and the historian notices... From 4000BC to 1500AD in about 1 minute, an entire game in about 10 (on my PC and a 50x80 map anyway).
If you want to actually see what's happening you can always press "y" to reveal the entire map (but that'll go much much slower).
As for the "fully automating", you'd still have to press down enter all this time for each turn at least (just keep it pressed down, and release it sometimes when a pop-up won't go away). So you can't really go away and come back 10 minutes later... But then, 10 minutes isn't too long, is it?